Ivancevic_Applied-Diff-Geom

914 Applied Differential Geometry: A Modern Introductionis the g l −valued curvature form on P (see the expression (5.333) below).In particular, a principal connection is flat iff its strength vanishes.5.11.2 Associated BundlesGiven a principal G−bundle π P : P → Q, let V be a manifold providedwith an effective left action G × V → V, (g, v) ↦→ gv of the Lie groupG. Let us consider the quotientY = (P × V )/G (5.330)of the product P × V by identification of elements (p, v) and (pg, g −1 v) forall g ∈ G. We will use the notation (pG, G −1 v) for its points. Let [p] denotethe restriction of the canonical surjectionP × V → (P × V )/G (5.331)to the subset {p} × V so that [p](v) = [pg](g −1 v). Then the map Y → Q,[p](V ) ↦→ π P (p), makes the quotient Y (5.330) to a fibre bundle over Q.Let us note that, for any G−bundle, there exists an associated principalG−bundle [Steenrod (1951)]. The peculiarity of the G−bundle Y (5.330)is that it appears canonically associated to a principal bundle P . Indeed,every bundle atlas Ψ P = {(U α , z α )} of P determines a unique associatedbundle atlasΨ = {(U α , ψ α (q) = [z α (q)] −1 )}of the quotient Y (5.330), and each automorphism of P also induces thecorresponding automorphism (5.346) of Y .Every principal connection A on a principal bundle P → Q inducesa unique connection on the associated fibre bundle Y (5.330). Given thehorizontal splitting of the tangent bundle T P relative to A, the tangent mapto the canonical map (5.331) defines the horizontal splitting of the tangentbundle T Y of Y and, consequently, a connection on Y → Q [Kobayashi andNomizu (1963/9)]. This is called the associated principal connection or aprincipal connection on a P −associated bundle Y → Q. If Y is a vectorbundle, this connection takes the formA = dq α ⊗ (∂ α − A p αI pij y j ∂ i ), (5.332)where I p are generators of the linear representation of the Lie algebra g r in